>Is every infinite subset S of omega_0 with the inherited order,>order isomorphic to omega_0?>>Yes. S is an ordinal, a denumerable ordinal.

???

>Let eta be the order type of S.>>Since S is a subset of omega_0, eta <= omega_0.>Since omega_0 is the smallest infinite ordinal, omega_0 <= eta.>Thus S and omega_0 are order isomorphic.>>Does the same reasoning hold to show that an uncountable subset>of omega_1 with the inherited order is order isomorphic to omega_1.>>It seems intuitive that since S is a subset of omega_1, that>order type S = eta <= omega_1. How could that be rigorously>shown?

Have you thought about this even a little bit?

Let s_0 be the smallest element of S. Let s_1 be thesmallest element of S minus s_0. Etc. Since S is infinite,you never run out of elements, you can always finds_{n+1}. It's easy to show that S = {s_n}:Say s is in S. Since s has only finitely many predecessors,there must be a stage in the construction when s isthe smalllest remaining element of S, and it becomess_n at that point.